A sharp bound for the functional calculus of $ρ$-contractions
Felix L. Schwenninger, Jens de Vries
TL;DR
This paper proves a sharp bound for the functional calculus of ρ-contractions: for a ρ-contraction A and any rational f mapping the closed unit disk to itself, the operator norm of f(A) is bounded by k_ρ(|f(0)|), where k_ρ is the decreasing function with k_ρ(0)=ρ and k_ρ(1)=1 defined by k_ρ(s) = (ρ/2)(1 - s^2) + sqrt((ρ^2/4)(1 - s^2)^2 + s^2). The authors obtain this bound via a new characterization of ρ-contractions and a Berger–Stampfli-type reduction to Blaschke factors, together with an integral representation of the analytic functional calculus. The result sharpens Okubo–Ando’s estimate for general ρ and recovers von Neumann’s inequality at ρ = 1 and Drury’s bound at ρ = 2, while establishing sharpness through a 2×2 example and Blaschke factors. This advances the quantitative understanding of how rational functional calculi behave on ρ-contractions and provides a unified, sharp bound across all ρ ≥ 1.
Abstract
Let $A$ be a $ρ$-contraction and $f$ a rational function mapping the closed unit disk into itself. With a new characterization of $ρ$-contractions we prove that \begin{align*} \big\|f(A)\big\|\leq \fracρ{2}\big(1-|f(0)|^{2}\big)+\sqrt{\frac{ρ^{2}}{4}\big(1-|f(0)|^{2}\big){}^{2}+|f(0)|^{2}}. \end{align*} We further show that this bound is sharp. This refines an estimate by Okubo--Ando and, for $ρ=2$, is consistent with a result by Drury.